So I have a cost function f(x1, x2, x3,....,xN) when x1, x2,...xN are matrices (so a function of images). I want to optmize this function using Polak Ribiere method. Since I know the general form of this cost function, I can write gradient as a function of x1,x2,x3,....xN.
While finding out the appropriate step size that satisfies the Wolfe conditions, I need to find out the norm of gradient and compare it with some value. I don't know what would be the norm of such a gradient which is a function of matrices. I can find out the squared sum of the individual gradient components but that itself would be a matrix. Do I compare element wise after this step ? But element wise comparison won't yield true until all the values are true, so this didn't make sense to me.
For cost function, it sounded reasonable to take pixelwise cost sum as the total cost, so I took that as the scalar equivalent of cost function. But can I do the same for gradient as well ?
Pardon me if I'm not rigorous enough.